Quantum Ruijsenaars system and quantum symmetric spaces
Abstract:
The aim of the talk is to present in an explicit form the radial components
of Casimir elements of Uq(gl(n)) corresponding
to the quantum analogues of homogeneous spaces GL(n)/SO(n)
and GL(2n)/Sp(n). These radial components lead
to commuting set of Hamiltonians of the quantum trigonometric n-particle
Ruijsenaars model. Explicit formulas for invariant vectors with respect
to a q-deformation of U(so(n)) in representation spaces
of
Uq(gl(n)) are presented. They are used to obtain
common eigenfunctions (which are, in fact, Macdonald polynomials) of commuting
set of Hamiltonians of Ruijsenaars model.